Английская Википедия:Densely defined operator
Шаблон:Short description In mathematics – specifically, in operator theory – a densely defined operator or partially defined operator is a type of partially defined function. In a topological sense, it is a linear operator that is defined "almost everywhere". Densely defined operators often arise in functional analysis as operations that one would like to apply to a larger class of objects than those for which they a priori "make sense".
Definition
A densely defined linear operator <math>T</math> from one topological vector space, <math>X,</math> to another one, <math>Y,</math> is a linear operator that is defined on a dense linear subspace <math>\operatorname{dom}(T)</math> of <math>X</math> and takes values in <math>Y,</math> written <math>T : \operatorname{dom}(T) \subseteq X \to Y.</math> Sometimes this is abbreviated as <math>T : X \to Y</math> when the context makes it clear that <math>X</math> might not be the set-theoretic domain of <math>T.</math>
Examples
Consider the space <math>C^0([0, 1]; \R)</math> of all real-valued, continuous functions defined on the unit interval; let <math>C^1([0, 1]; \R)</math> denote the subspace consisting of all continuously differentiable functions. Equip <math>C^0([0, 1]; \R)</math> with the supremum norm <math>\|\,\cdot\,\|_\infty</math>; this makes <math>C^0([0, 1]; \R)</math> into a real Banach space. The differentiation operator <math>D</math> given by <math display=block>(\mathrm{D} u)(x) = u'(x)</math> is a densely defined operator from <math>C^0([0, 1]; \R)</math> to itself, defined on the dense subspace <math>C^1([0, 1]; \R).</math> The operator <math>\mathrm{D}</math> is an example of an unbounded linear operator, since <math display=block>u_n (x) = e^{- n x} \quad \text{ has } \quad \frac{\left\|\mathrm{D} u_n\right\|_{\infty}}{\left\|u_n\right\|_\infty} = n.</math> This unboundedness causes problems if one wishes to somehow continuously extend the differentiation operator <math>D</math> to the whole of <math>C^0([0, 1]; \R).</math>
The Paley–Wiener integral, on the other hand, is an example of a continuous extension of a densely defined operator. In any abstract Wiener space <math>i : H \to E</math> with adjoint <math>j := i^* : E^* \to H,</math> there is a natural continuous linear operator (in fact it is the inclusion, and is an isometry) from <math>j\left(E^*\right)</math> to <math>L^2(E, \gamma; \R),</math> under which <math>j(f) \in j\left(E^*\right) \subseteq H</math> goes to the equivalence class <math>[f]</math> of <math>f</math> in <math>L^2(E, \gamma; \R).</math> It can be shown that <math>j\left(E^*\right)</math> is dense in <math>H.</math> Since the above inclusion is continuous, there is a unique continuous linear extension <math>I : H \to L^2(E, \gamma; \R)</math> of the inclusion <math>j\left(E^*\right) \to L^2(E, \gamma; \R)</math> to the whole of <math>H.</math> This extension is the Paley–Wiener map.
See also
References
Шаблон:Hilbert space Шаблон:Banach spaces Шаблон:Functional analysis
